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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p><dfn class="terminology">Diagonalization</dfn> Consider</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\bf x}^{\prime}={\bf A}\,{\bf x}+{\bf g}(t),\quad \alpha \leq t \leq \beta,
\end{equation*}
</div>
<p class="continuation">where <span class="process-math">\({\bf A}\)</span> is a constant coefficient matrix. Introduce <span class="process-math">\({\bf x}={\bf T}\, \vec{y}\text{,}\)</span> where <span class="process-math">\({\bf T}=(\vec{\xi}^{(1)}, \vec{\xi}^{(2)}, \cdots, \vec{\xi}^{(n)})\)</span> and <span class="process-math">\(\vec{\xi}^{(1)}, \vec{\xi}^{(2)}, \cdots, \vec{\xi}^{(n)}\)</span> are eigenvectors corresponding to eigenvalues <span class="process-math">\(r_1, r_2, \cdots, r_n\text{.}\)</span> Then</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\bf T} \,\vec{y}^{\prime}={\bf A}\,{\bf T}\, \vec{y}+{\bf g}(t) \to
\vec{y}^{\prime}={\bf T}^{-1}\,{\bf A}\,{\bf T}\, \vec{y}+{\bf T}^{-1}\,{\bf g}(t) =
\left(
\begin{array}{cccc}
r_1 &amp; 0 &amp; \cdots &amp; 0\\
0&amp;r_2&amp;\cdots&amp;0\\
\vdots &amp; \vdots &amp; \ddots &amp; \vdots\\
0 &amp; 0&amp; \vdots &amp; r_n
\end{array}
\right)\vec{y}+\left(
\begin{array}{c}
h_1(t)\\
h_2(t)\\
\vdots\\
h_n(t)
\end{array}
\right).
\end{equation*}
</div>
<p class="continuation">In component form</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\begin{array}{c}
y_1^{\prime}=r_1 y_1+h_1(t)\\
y_2^{\prime}=r_2 y_2+h_2(t)\\
\vdots\\
y_n^{\prime}=r_n y_n+h_n(t)\\
\end{array}
\end{equation*}
</div>
<p class="continuation">which are decoupled equations. Using the method of integrating factor, we have solutions</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\begin{array}{c}
y_1=C_1 e^{r_1 t}+e^{r_1 t} \int e^{-r_1 t} h_1(t) \mathrm{d} t,\\
y_2=C_2 e^{r_2 t}+e^{r_2 t} \int e^{-r_2 t} h_2(t) \mathrm{d} t,\\
\vdots\\
y_n=C_n e^{r_n t}+e^{r_n t} \int e^{-r_n t} h_n(t) \mathrm{d} t.\\
\end{array}
\end{equation*}
</div>
<p class="continuation">The solution is then</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\bf x}={\bf T}\,\vec{y}.
\end{equation*}
</div>
<span class="incontext"><a href="sec6_6.html#p-292" class="internal">in-context</a></span>
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